Implementing GARCH(2,2) QMLE: where does the data (squared returns) come into play?

Question

I am trying to implement a QMLE estimation of GARCH(2,2) model as a side project.

We can represent GARCH(2,2) as follows: \begin{aligned} r_{t} &= \mu_{t} + \epsilon_{t}, \\ \mu_{t} &= 0, \\ \epsilon_{t} &= \sigma_{t}z_{t}, \quad z_{t} \sim N(0,1), \\ \sigma_{t+1}^{2} &= \omega + \alpha_{1}\epsilon_{t}^{2} + \beta_{1}\sigma_{t}^{2} + \alpha_{2}\epsilon_{t-1}^{2} + \beta_{2}\sigma_{t-1}^{2}. \end{aligned} Using standard arguments, we can write the log-likelihood as $$ \sum_{t}^{T}l(\theta|r_{t}) = \sum_{t=1}^{T}\log\bigg[ \frac{1}{\sqrt{2\pi\sigma^{2}_{t}(\theta)}}\exp\{\frac{r_{t}^{2}}{2\sigma(\theta)_{t}^{2}}\}\bigg] $$ where $\theta$ is just the vector of parameters.

This would be estimated, at each time $t$ by plugging in $\omega + \alpha_{1}\epsilon_{t-1}^{2} + \beta_{1}\sigma_{t-1}^{2} + \alpha_{2}\epsilon_{t-2}^{2} + \beta_{2}\sigma_{t-2}^{2}$ instead of $\sigma_{t}^{2}$ and $r_{t}^{2}$ (data). However, my problem is basically that I cannot understand whether

1. $r_{t}^{2}$ is the data that we have (which would be logical) or
2. if $r_{t}^{2}$ is simply generated from previous iterations of $\sigma_{t}^{2}$, as $r_{t}^{2} = \epsilon_{t}^{2} = z_{t}^{2}\sigma_{t}^{2}=z_{t}^{2}(\omega + \alpha_{1}\epsilon_{t-1}^{2} + \beta_{1}\sigma_{t-1}^{2} + \alpha_{2}\epsilon_{t-2}^{2} + \beta_{2}\sigma_{t-2}^{2})$.

The latter doesn't make sense, and it means I only need to initialize the process and then I wouldn't need any data to fit it, and this is exactly what confuses me.

If the former is true, then how can we say that $r_{t} = \epsilon_{t}$?

I have already checked this post, which was quite helpful, but still does not answer my question.

Perhaps it has to something with conditioning on information at time $t$, but I am struggling to clearly understand this.

Answer 1

If the model were true, you would find that $$ r_{t}^{2} = \epsilon_{t}^{2} = z_{t}^{2}\sigma_{t}^{2}=z_{t}^{2}(\omega + \alpha_{1}\epsilon_{t-1}^{2} + \beta_{1}\sigma_{t-1}^{2} + \alpha_{2}\epsilon_{t-2}^{2} + \beta_{2}\sigma_{t-2}^{2}). $$ This provides a way to generate a time series $\{r_t\}_{t=1}^T$ given the starting values and the parameters. Suppose that God or nature did that, and now you have observed $\{r_t\}_{t=1}^T$. If you wish to find out what parameters and starting values were used to generate this time series, you can use the data $\{r_t\}_{t=1}^T$ to estimate them using, say, maximum likelihood.

There is no contradiction between how the data were generated and that you might want to find the parameters that you cannot observe directly.

P.S. Note that $z_t\sim \text{i.i.d.}\ N(0,1)$, not only $z_t\sim N(0,1)$.